Questions: Graph the function. r(x) = (x-1)/(2x-5)

Transcript text: Graph the function. \[ r(x)=\frac{x-1}{2 x-5} \]

Solution

Solution Steps

Step 1: Identify the Asymptotes

The function given is $ r(x) = \frac{x-1}{2x-5} $.

Vertical Asymptote: This occurs where the denominator is zero. Set $ 2x - 5 = 0 $ and solve for $ x $. \[ 2x - 5 = 0 \implies x = \frac{5}{2} \]

Step 2: Determine the Horizontal Asymptote

For rational functions, if the degree of the numerator and the denominator are the same, the horizontal asymptote is the ratio of the leading coefficients.

The leading coefficient of the numerator is 1.
The leading coefficient of the denominator is 2.

Thus, the horizontal asymptote is: \[ y = \frac{1}{2} \]

Step 3: Analyze the Function

The function $ r(x) = \frac{x-1}{2x-5} $ is a rational function with a vertical asymptote at $ x = \frac{5}{2} $ and a horizontal asymptote at $ y = \frac{1}{2} $.

Final Answer

The function $ r(x) = \frac{x-1}{2x-5} $ has a vertical asymptote at $ x = \frac{5}{2} $ and a horizontal asymptote at $ y = \frac{1}{2} $.

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 10}, "commands": ["y = (x-1)/(2x-5)"], "latex_expressions": ["$y = \\frac{x-1}{2x-5}$"]}

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